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The Stone–von Neumann theorem is a fundamental result which unified the competing quantum-mechanical models of matrix mechanics and wave mechanics. In this article, we continue the broad generalization set out by Huang and Ismert and by Hall, Huang, and Quigg, analyzing representations of locally compact quantum-dynamical systems defined on Hilbert modules, of which the classical result is a special case. We introduce a pair of modular representations which subsume numerous models available in the literature and, using the classical strategy of Rieffel, prove a Stone–von Neumann-type theorem for maximal actions of regular locally compact quantum groups on elementary C*-algebras. In particular, we generalize the Mackey–Stone–von Neumann theorem to regular locally compact quantum groups whose trivial actions on C are maximal and recover the multiplicity results of Hall, Huang, and Quigg. With this characterization in hand, we prove our main result showing that if a dynamical system (G , A , α) satisfies the multiplicity assumption of the generalized Stone–von Neumann theorem, and if the coefficient algebra A admits a faithful state, then the spectrum of the iterated crossed product G ^ op ⋉ (G ⋉ A) consists of a single point. In the case of a separable coefficient algebra or a regular acting quantum group, we further characterize features of this system, and thus obtain a partial converse to the Stone–von Neumann theorem in the quantum group setting. As a corollary, we show that a regular locally compact quantum group satisfies the generalized Stone–von Neumann theorem if and only if it is strongly regular. [ABSTRACT FROM AUTHOR]

We construct a quantum extension of the classical Jacobi semigroup to the non-commutative von Neumann algebra B (L 2 ((- 1 , 1) , (1 - x) α (1 + x) β d x) for certain values of α , β and study some of its properties. [ABSTRACT FROM AUTHOR]

We prove that a large class of nonamenable almost periodic type III 1 factors M, including all McDuff factors that tensorially absorb R ∞ and all free Araki–Woods factors, satisfy Haagerup–Størmer's conjecture (1988): any pointwise inner automorphism of M is the composition of an inner and a modular automorphism. [ABSTRACT FROM AUTHOR]

We revisit Fourier’s approach to solve the heat equation on the circle in the context of (twisted) reduced group C*-algebras, convergence of Fourier series and semigroups associated to negative definite functions. We introduce some heat properties for countably infinite groups and investigate when they are satisfied. Kazhdan’s property (T) is an obstruction to the weakest property, and our findings leave open the possibility that this might be the only one. On the other hand, many groups with the Haagerup property satisfy the strongest version. We show that this heat property implies that the associated heat problem has a unique solution regardless of the choice of the initial datum. [ABSTRACT FROM AUTHOR]

Let X be a vector field and Y be a co-vector field on a smooth manifold M. Does there exist a smooth Riemannian metric g α β on M such that Y β = g α β X α ? The main result of this note gives necessary and sufficient conditions for this to be true. As an application of this result we provide a gradient-flow characterisation for dissipative quantum systems. Namely, we show that finite-dimensional ergodic Lindblad equations admit a gradient flow structure for the von Neumann relative entropy if and only if the condition of bkm-detailed balance holds. [ABSTRACT FROM AUTHOR]

We present some more foundations for a theory of real structure in operator spaces and algebras, in particular concerning the real case of the theory of injectivity, and the injective, ternary, and C ∗ -envelope. We consider the interaction between these topics and the complexification. We also generalize many of these results to the setting of operator spaces and systems acted upon by a group. [ABSTRACT FROM AUTHOR]

We study the gauge-invariant ideal structure of the Nica-Toeplitz algebra N T (X) of a product system (A, X) over N n . We obtain a clear description of X-invariant ideals in A, that is, restrictions of gauge-invariant ideals in N T (X) to A. The main result is a classification of gauge-invariant ideals in N T (X) for a proper product system in terms of families of ideals in A. We also apply our results to higher-rank graphs. [ABSTRACT FROM AUTHOR]

In this paper, we study the Haagerup property of semigroup crossed products for lattice ordered groups. Let G be a lattice ordered group. We show that the reduced semigroup C*-algebra C*r (G+) has the Haagerup property if and only if G has the Haagerup property. Let G act on a unital C*-algebra through an action α, we show that if the reduced semigroup crossed product has the Haagerup property, then both and G have the Haagerup property. On the other hand, it is shown that if G is s-amenable and has the Haagerup property, then has the Haagerup property. [ABSTRACT FROM AUTHOR]

It is well-known that the maximalization of a coaction of a locally compact group on a C*-algebra enjoys a universal property. We show how this important property can be deduced from a categorical framework by exploiting certain properties of the maximalization functor for coactions. We also provide a dual proof for the universal property of normalization of coactions. [ABSTRACT FROM AUTHOR]

We show the reduced C ∗ -algebra of a graded ample groupoid is a strongly graded C ∗ -algebra if and only if the corresponding Steinberg algebra is a strongly graded ring. We use this to show that the graph C ∗ -algebra of a countable directed graph is a strongly Z -graded C ∗ -algebra if and only if the Leavitt path algebra is a strongly Z -graded ring. The latter of these was shown previously to be equivalent to the graph satisfying property (Y). [ABSTRACT FROM AUTHOR]

If I is a (two-sided) ideal of a ring R, we let ann l (I) = { r ∈ R ∣ r I = 0 } , ann r (I) = { r ∈ R ∣ I r = 0 } , and ann (I) = ann l (I) ∩ ann r (I) be the left, the right and the double annihilators. An ideal I is said to be an annihilator ideal if I = ann (J) for some ideal J (equivalently, ann (ann (I)) = I ). We study annihilator ideals of Leavitt path algebras and graph C ∗ -algebras. Let L K (E) be the Leavitt path algebra of a graph E over a field K. If I is an ideal of L K (E) , it has recently been shown that ann (I) is a graded ideal (with respect to the natural grading of L K (E) by Z ). We note that ann l (I) and ann r (I) are also graded. If I is graded, we show that ann l (I) = ann r (I) = ann (I) and describe ann (I) in terms of the properties of a pair of sets of vertices of E, known as an admissible pair, which naturally corresponds to I. Using such a description, we present properties of E which are equivalent with the requirement that each graded ideal of L K (E) is an annihilator ideal. We show that the same properties of E are also equivalent with each of the following conditions: (1) the lattice of graded ideals of L K (E) is a Boolean algebra; (2) each closed gauge-invariant ideal of C ∗ (E) is an annihilator ideal; (3) the lattice of closed gauge-invariant ideals of C ∗ (E) is a Boolean algebra. In addition, we present properties of E which are equivalent with each of the following conditions: (1) each ideal of L K (E) is an annihilator ideal; (2) the lattice of ideals of L K (E) is a Boolean algebra; (3) each closed ideal of C ∗ (E) is an annihilator ideal; (4) the lattice of closed ideals of C ∗ (E) is a Boolean algebra. [ABSTRACT FROM AUTHOR]

We give a survey of Rokhlin-type properties of group actions on operator algebras. Though we briefly review the classification of group actions on von Neumann algebras, we mainly overview group actions on C ∗ -algebras, focusing on classification of group actions and preservation of properties under taking C ∗ -crossed products for actions with Rokhlin-type properties. [ABSTRACT FROM AUTHOR]

Let A be an algebraically simple, separable, nuclear, Z -stable C ∗ -algebra for which the trace space T(A) is a Bauer simplex and the extremal boundary ∂ e T (A) has finite covering dimension. We prove that each automorphism α on A is cocycle conjugate to its tensor product with the trivial automorphism on the Jiang–Su algebra. At least for single automorphisms this generalizes a recent result by Gardella–Hirshberg–Vaccaro. If α is strongly outer as an action of Z , we prove it has finite Rokhlin dimension with commuting towers. As a consequence it tensorially absorbs any automorphism on the Jiang–Su algebra. [ABSTRACT FROM AUTHOR]

It is introduced an analogue of the orbit-breaking subalgebra for the case of free flows on locally compact metric spaces, which has a natural approximate structure in terms of a fixed point and any nested sequence of central slices around this point. It is shown that in the case of minimal flows admitting a compact Cantor central slice, the resulting C ∗ -algebra is the stabilization of the Putnam orbit-breaking subalgebra associated to the induced homeomorphism on the central slice. This construction provides an alternative characterization (up to stabilization) of the orbit-breaking subalgebra introduced by Putnam for minimal homeomorphisms of Cantor spaces in terms of suspension flows associated to such dynamical systems. [ABSTRACT FROM AUTHOR]

We provide a dynamical proof of the van der Corput inequality for sequences in Hilbert spaces that is based on the Furstenberg correspondence principle. This is done by reducing the inequality to the mean ergodic theorem for contractions on Hilbert spaces. The key difficulty therein is that the Furstenberg correspondence principle is, a priori, limited to scalar-valued sequences. We, therefore, discuss how interpreting the Furstenberg correspondence principle via the Gelfand–Naimark–Segal construction for C ∗ -algebras allows to study not just scalar but general Hilbert space-valued sequences in terms of unitary operators. This yields a proof of the van der Corput inequality in the spirit of the Furstenberg correspondence principle and the flexibility of this method is discussed via new proofs for different variants of the inequality. [ABSTRACT FROM AUTHOR]

We consider groupoids constructed from a finite number of commuting local homeomorphisms acting on a compact metric space and study generalized Ruelle operators and $ C^{\ast } $ -algebras associated to these groupoids. We provide a new characterization of $ 1 $ -cocycles on these groupoids taking values in a locally compact abelian group, given in terms of $ k $ -tuples of continuous functions on the unit space satisfying certain canonical identities. Using this, we develop an extended Ruelle–Perron–Frobenius theory for dynamical systems of several commuting operators ( $ k $ -Ruelle triples and commuting Ruelle operators). Results on KMS states on $ C^{\ast } $ -algebras constructed from these groupoids are derived. When the groupoids being studied come from higher-rank graphs, our results recover existence and uniqueness results for KMS states associated to the graphs. [ABSTRACT FROM AUTHOR]

We introduce Poisson boundaries of II $_1$ factors with respect to density operators that give the traces. The Poisson boundary is a von Neumann algebra that contains the II $_1$ factor and is a particular example of the boundary of a unital completely positive map as introduced by Izumi. Studying the inclusion of the II $_1$ factor into its boundary, we develop a number of notions, such as double ergodicity and entropy, that can be seen as natural analogues of results regarding the Poisson boundaries introduced by Furstenberg. We use the techniques developed to answer a problem of Popa by showing that all finite factors satisfy his MV property. We also extend a result of Nevo by showing that property (T) factors give rise to an entropy gap. [ABSTRACT FROM AUTHOR]

In this paper, we construct uncountably many examples of multiparameter CCR flows, which are not pullbacks of $1$ -parameter CCR flows, with any given index. Moreover, the constructed CCR flows are type I in the sense that the associated product system is the smallest subsystem containing its units. [ABSTRACT FROM AUTHOR]

We establish several new characterizations of amenable $W^*$ - and $C^*$ -dynamical systems over arbitrary locally compact groups. In the $W^*$ -setting we show that amenability is equivalent to (1) a Reiter property and (2) the existence of a certain net of completely positive Herz–Schur multipliers of $(M,G,\alpha)$ converging point weak* to the identity of $G\bar {\ltimes }M$. In the $C^*$ -setting, we prove that amenability of $(A,G,\alpha)$ is equivalent to an analogous Herz–Schur multiplier approximation of the identity of the reduced crossed product $G\ltimes A$ , as well as a particular case of the positive weak approximation property of Bédos and Conti [On discrete twisted $C^*$ -dynamical systems, Hilbert $C^*$ -modules and regularity. Münster J. Math.5 (2012), 183–208] (generalized to the locally compact setting). When $Z(A^{**})=Z(A)^{**}$ , it follows that amenability is equivalent to the 1-positive approximation property of Exel and Ng [Approximation property of $C^*$ -algebraic bundles. Math. Proc. Cambridge Philos. Soc.132(3) (2002), 509–522]. In particular, when $A=C_0(X)$ is commutative, amenability of $(C_0(X),G,\alpha)$ coincides with topological amenability of the G-space $(G,X)$. [ABSTRACT FROM AUTHOR]

We characterize topological conjugacy classes of one-sided topological Markov shifts in terms of the associated Cuntz–Krieger algebras and their gauge actions with potentials. [ABSTRACT FROM AUTHOR]

Suppose that (G , G +) is a lattice-ordered abelian group with positive cone G + . Adji and Raeburn have shown a result about the structure of the primitive ideal space of the C ∗ -algebra B G + × α G + for a totally ordered abelian group (Adji and Raeburn in Integral Equ Oper Theory 48:281–293, 2004, Theorem 3.1). In this paper, we show if ∑ (G) is the chain of subgroups H of G, where H : = H + - H + and H + is any hereditary subsemigroup of G + . Then, there exists a well-defined map F from the disjoint union ⨆ { H ^ : H ∈ ∑ G } to the primitive ideals of the C ∗ -algebra B G + × α G + . Our result is interesting and has more challenges, since we are working under the assumption that the group G is lattice-ordered and extends their result to more general cases. [ABSTRACT FROM AUTHOR]

We generalize Condition (K) from directed graphs to Boolean dynamical systems and show that a locally finite Boolean dynamical system $({{\mathcal {B}}},{{\mathcal {L}}},\theta)$ with countable ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ satisfies Condition (K) if and only if every ideal of its $C^*$ -algebra is gauge-invariant, if and only if its $C^*$ -algebra has the (weak) ideal property, and if and only if its $C^*$ -algebra has topological dimension zero. As a corollary we prove that if the $C^*$ -algebra of a locally finite Boolean dynamical system with ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$ countable either has real rank zero or is purely infinite, then $({{\mathcal {B}}}, {{\mathcal {L}}}, \theta)$ satisfies Condition (K). We also generalize the notion of maximal tails from directed graph to Boolean dynamical systems and use this to give a complete description of the primitive ideal space of the $C^*$ -algebra of a locally finite Boolean dynamical system that satisfies Condition (K) and has countable ${{\mathcal {B}}}$ and ${{\mathcal {L}}}$. [ABSTRACT FROM AUTHOR]

This paper is a continuation of the paper, Matsumoto ['Subshifts, $\lambda $ -graph bisystems and $C^*$ -algebras', J. Math. Anal. Appl. 485 (2020), 123843]. A $\lambda $ -graph bisystem consists of a pair of two labeled Bratteli diagrams satisfying a certain compatibility condition on their edge labeling. For any two-sided subshift $\Lambda $ , there exists a $\lambda $ -graph bisystem satisfying a special property called the follower–predecessor compatibility condition. We construct an AF-algebra ${\mathcal {F}}_{\mathcal {L}}$ with shift automorphism $\rho _{\mathcal {L}}$ from a $\lambda $ -graph bisystem $({\mathcal {L}}^-,{\mathcal {L}}^+)$ , and define a $C^*$ -algebra ${\mathcal R}_{\mathcal {L}}$ by the crossed product. It is a two-sided subshift analogue of asymptotic Ruelle algebras constructed from Smale spaces. If $\lambda $ -graph bisystems come from two-sided subshifts, these $C^*$ -algebras are proved to be invariant under topological conjugacy of the underlying subshifts. We present a simplicity condition of the $C^*$ -algebra ${\mathcal R}_{\mathcal {L}}$ and the K-theory formulas of the $C^*$ -algebras ${\mathcal {F}}_{\mathcal {L}}$ and ${\mathcal R}_{\mathcal {L}}$. The K-group for the AF-algebra ${\mathcal {F}}_{\mathcal {L}}$ is regarded as a two-sided extension of the dimension group of subshifts. [ABSTRACT FROM AUTHOR]

Let P ⋊ N × be a semidirect product of the additive semigroup P = { 0 , 2 , 3 , ⋯ } by a multiplicative positive natural numbers semigroup N ×. A covariant isometric representation of a quasi-lattice ordered group was introduced in order to resolve the uniqueness property of a semigroup C ∗ -algebra. The concept of the covariant isometric representation for a quasi-lattice ordered semigroup is extended to a non quasi-lattice ordered semigroup P ⋊ N × in this paper. We construct a generalized Toeplitz algebra T (P ⋊ N ×) with the universal property of covariant isometric representations of P ⋊ N × and prove that the generalized Toeplitz algebra T (P ⋊ N ×) is the algebraic universal C ∗ -algebra of the relations of its generators. We also show that there don’t exist the KMS β states at the inverse temperature 0 < β < 1 and compute the KMS β states for 1 ≤ β < ∞ of the natural C ∗ -dynamical system of T (P ⋊ N ×) . [ABSTRACT FROM AUTHOR]

Recent work by Baum et al. ['Expanders, exact crossed products, and the Baum–Connes conjecture', Ann. K-Theory 1(2) (2016), 155–208], further developed by Buss et al. ['Exotic crossed products and the Baum–Connes conjecture', J. reine angew. Math. 740 (2018), 111–159], introduced a crossed-product functor that involves tensoring an action with a fixed action $(C,\unicode[STIX]{x1D6FE})$ , then forming the image inside the crossed product of the maximal-tensor-product action. For discrete groups, we give an analogue for coaction functors. We prove that composing our tensor-product coaction functor with the full crossed product of an action reproduces their tensor-crossed-product functor. We prove that every such tensor-product coaction functor is exact, and if $(C,\unicode[STIX]{x1D6FE})$ is the action by translation on $\ell ^{\infty }(G)$ , we prove that the associated tensor-product coaction functor is minimal, thereby recovering the analogous result by the above authors. Finally, we discuss the connection with the $E$ -ization functor we defined earlier, where $E$ is a large ideal of $B(G)$. [ABSTRACT FROM AUTHOR]

In this paper we construct a quantum extension of the Laguerre semigroup and study its properties. In particular we show that it has a unique pure invariant state and any initial state converges to this invariant state. For initial states satisfying a finite energy condition, convergence is exponentially fast. [ABSTRACT FROM AUTHOR]

Let G be a locally compact group that acts on a C ∗ -algebra A through an action β. Generalizing an existing result for discrete group actions, we show that if there exist a tracial state ϕ on the full crossed product A ⋊ β G and a positive element x 0 in the center of A ⋊ β G with ϕ (x 0) ≠ 0 , then A ⋊ β G has strong property T if and only if G has property T and (A ⋊ β G , A) has strong property T. Furthermore, generalizing another existing result concerning property T for reduced group C ∗ -algebras, we show that if there exist a β-invariant tracial state τ on A and a continuous function f : G → A with compact support such that ∫ G τ (f 2 (s)) d s ≠ 0 , then the reduced crossed product A ⋊ β , r G has strong property T if and only if G has property T and (A ⋊ β , r G , A) has strong property T. [ABSTRACT FROM AUTHOR]

Using the recently developed theory of Herz–Schur multipliers of a C ∗ -dynamical system, we prove equality of the Haagerup constants for a C ∗ -algebra and its crossed product by an amenable action. [ABSTRACT FROM AUTHOR]

Let $G$ be a second countable locally compact Hausdorff topological group and $P$ be a closed subsemigroup of $G$ containing the identity element $e\in G$. Assume that the interior of $P$ is dense in $P$. Let $\unicode[STIX]{x1D6FC}=\{{\unicode[STIX]{x1D6FC}_{x}\}}_{x\in P}$ be a semigroup of unital normal $\ast$ -endomorphisms of a von Neumann algebra $M$ with separable predual satisfying a natural measurability hypothesis. We show that $\unicode[STIX]{x1D6FC}$ is an $E_{0}$ -semigroup over $P$ on $M$. [ABSTRACT FROM AUTHOR]

First, we generalize the definition of a locally compact topology given by Paterson and Welch for a sequence of locally compact spaces to the case where the underlying spaces are $T_{1}$ and sober. We then consider a certain semilattice of basic open sets for this topology on the space of all paths on a graph and impose relations motivated by the definitions of graph C*-algebra in order to recover the boundary path space of a graph. This is done using techniques of pointless topology. Finally, we generalize the results to the case of topological graphs. [ABSTRACT FROM AUTHOR]

A λ-graph bisystem L consists of a pair (L − , L +) of two labelled Bratteli diagrams, that presents a two-sided subshift Λ L . We will construct a compact totally disconnected metric space consisting of tilings of a two-dimensional half plane from a λ-graph bisystem. The tiling space has a certain AF-equivalence relation written R L with a natural shift homeomorphism σ L coming from the shift homeomorphism σ Λ L on the subshift Λ L . The equivalence relation R L yields an AF-algebra F L with an automorphism ρ L induced by σ L . We will study invariance of the étale equivalence relation R L , the groupoid G L = R L ⋊ σ L Z and the groupoid C ∗ -algebras C ∗ (R L ) , C ∗ (G L ) under topological conjugacy of the presenting two-sided subshifts. [ABSTRACT FROM AUTHOR]

Let $(G,\unicode[STIX]{x1D6EC})$ be a self-similar $k$ -graph with a possibly infinite vertex set $\unicode[STIX]{x1D6EC}^{0}$. We associate a universal C*-algebra ${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$ to $(G,\unicode[STIX]{x1D6EC})$. The main purpose of this paper is to investigate the ideal structures of ${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$. We prove that there exists a one-to-one correspondence between the set of all $G$ -hereditary and $G$ -saturated subsets of $\unicode[STIX]{x1D6EC}^{0}$ and the set of all gauge-invariant and diagonal-invariant ideals of ${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$. Under some conditions, we characterize all primitive ideals of ${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$. Moreover, we describe the Jacobson topology of some concrete examples, which includes the C*-algebra of the product of odometers. On the way to our main results, we study self-similar $P$ -graph C*-algebras in depth. [ABSTRACT FROM AUTHOR]

Following the operator algebraic approach to Gabor analysis, we construct frames of translates for the Hilbert space localisation of the Morita equivalence bimodule arising from a groupoid equivalence between Hausdorff groupoids, where one of the groupoids is étale and with a compact unit space. For finitely generated and projective submodules, we show these frames are orthonormal bases if and only if the module is free. We then apply this result to the study of localised Wannier bases of spectral subspaces of Schrödinger operators with atomic potentials supported on (aperiodic) Delone sets. The noncommutative Chern numbers provide a topological obstruction to fast-decaying Wannier bases and we show this result is stable under deformations of the underlying Delone set. [ABSTRACT FROM AUTHOR]

We show that, up to strong cocycle conjugacy, every countable exact group admits a unique equivariantly $\mathcal {O}_{2}$ -absorbing, pointwise outer action on the Cuntz algebra $\mathcal {O}_{2}$ with the quasi-central approximation property (QAP). In particular, we establish the equivariant analogue of the Kirchberg $\mathcal {O}_{2}$ -absorption theorem for these groups. [ABSTRACT FROM AUTHOR]

We prove that for any countable group $\unicode[STIX]{x1D6E4}$ , there exists a free minimal continuous action $\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D6E4}\curvearrowright {\mathcal{C}}$ on the Cantor set admitting an invariant Borel probability measure. [ABSTRACT FROM AUTHOR]

We define a tracial analog of the notion called a sequentially split ∗ -homomorphism between C ∗ -algebras due to Barlak and Szabó and show that several important approximation properties related to the classification theory of C ∗ -algebras pass from the target algebra to the domain algebra. Then, we show that this framework arises from tracial Rokhlin properties of a finite group action and an inclusion of unital C ∗ -algebras. [ABSTRACT FROM AUTHOR]

Let G be an infinite countable group and A be a finite set. If ∑ ⊆ AG is a strongly irreducible subshift of finite type, we endow a locally compact and Hausdorff topology on the homoclinic equivalence relation G on ∑ and show that the reduced C*-algebra C r * (G) of G is a unital simple approximately finite (AF)-dimensional C*-algebra. The shift action of G on ∑ induces a canonical automorphism action of G on the C*-algebra C r * (G) . We give the notion of noncommutative dynamical entropy invariants for amenable group actions on C*-algebras, and show that, if G is an amenable group, then the noncommutative topological entropy of the canonical automorphism action of G on C r * (G) is equal to the topology entropy of the shift action of G on ∑. We also establish the variational principle with respect to the noncommutative measure entropy and the topological entropy for the C*-dynamical system ( C r * (G) , G). [ABSTRACT FROM AUTHOR]

In the present paper, we study infinite dimensional orthogonality preserving the second-order nonlinear Markov operators. It is proved that subjectivity of the second-order nonlinear Markov operators is equivalent to the orthogonality preserves in the class of π-Volterra operators. Moreover, a full description of such kind of operators has been found in terms of heredity coefficients. Besides, we are able to represent these operators their canonical forms. Furthermore, some properties of orthogonality preserving the second-order nonlinear operators and their fixed points are studied. [ABSTRACT FROM AUTHOR]

We consider fibrewise singly generated Fell bundles over étale groupoids. Given a continuous real-valued 1-cocycle on the groupoid, there is a natural dynamics on the cross-sectional algebra of the Fell bundle. We study the Kubo–Martin–Schwinger equilibrium states for this dynamics. Following work of Neshveyev on equilibrium states on groupoid C*-algebras, we describe the equilibrium states of the cross-sectional algebra in terms of measurable fields of states on the C*-algebras of the restrictions of the Fell bundle to the isotropy subgroups of the groupoid. As a special case, we obtain a description of the trace space of the cross-sectional algebra. We apply our result to generalise Neshveyev's main theorem to twisted groupoid C*-algebras, and then apply this to twisted C*-algebras of strongly connected finite k-graphs. [ABSTRACT FROM AUTHOR]